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Numerical Analysis of a Finite Element/Volume Penalty Method 183
Remark 4 (Technical assumptions). Some assumptions we made are only technical
and can surely be relaxed without changing the convergence results. For
example, the inclusion, which we supposed circular, could be any smooth
domain. Note that a convex polygon is not acceptable, as it is seen by the
problem from the outside, so that u may no longer be in H 2 , which rules out
some of the approximation properties we made.
Remark 5 (Convergence in space). The poor rate of convergence in h is optimal
for a non-boundary-fitted mesh, at least if we consider the H 1 -error overall
Ω. Indeed, as the solution is constant inside O, non-constant outside with a
jump in the normal derivative, the error within each element intersecting ∂O
is a O(1) in this L ∞ -norm. By summing up over all those triangles, which
cover a zone whose measure scales like h, we end up with this h 1/2 -error. Note
that a better convergence could be expected, in theory, if one considers only
the error in the domain of interest Ω \ O, the question being now whether the
bad convergence in the neighborhood of ∂O pollutes the overall approximation.
Our feeling is that this pollution actually occurs, because nothing is done
in the present approach to distinguish the real domain of interest from the
fictitious domain (inside the obstacle), so that the method tends to balance
the errors on both sides. An interesting way to privilegiate the side of interest
is proposed in [DP02] for a boundary penalty method; it consists in having
the diffusion coefficient vanish within Ω. Note that other methods have been
proposed to reach the optimal convergence rate on non-boundary-fitted mesh
(see [Mau01]), but they are less straightforward to implement.
The simplest way to improve the actual order of convergence is to carry
out a local refinement strategy in the neighborhood of ∂O (see [RAB06], for
example). By using elements of scale h 2 in this zone, one recovers the first
order convergence in space, at least in practice.
Remark 6 (Meaning of λ). As we already mentioned, the Lagrange multiplier
λ can often be interpreted as a force or a heat source which ensures the
prescribed constraint, depending on the context, and it may be useful to
estimate this term with accuracy. For example, the problem we considered
can be reformulated as a control problem: find a source term g with zero
mean value (no heat is injected into the system) which is subject to vanish
outside O, such that the solution u to
−△u = f + g, u =0 on∂Ω,
is constant over O. This equation is to be considered in the distributional
sense, as g is surely not a function. (If it were L 2 , for example, u would be
in H 2 (Ω), which is surely not true as its normal derivative overcomes a jump
through ∂O.) Abstractly speaking, this source term g is simply the opposite
of the linear functional ξ which we introduced (see (4)) and it is related to
the Lagrange multiplier λ (see (5))
∫
∫
〈g, v〉 = −〈ξ,v〉 = − λ ·∇v = − λ · n v.
O
∂O